PHYSICAL REVIEW E 92, 042129 (2015)
Quasi-steady-state analysis of coupled flashing ratchets
Ethan Levien and Paul C. Bressloff*
Department of Mathematics, University of Utah, 155 South 1400 East, Salt Lake City, Utah 84112, USA
(Received 29 May 2015; revised manuscript received 4 September 2015; published 12 October 2015)
We perform a quasi-steady-state (QSS) reduction of a flashing ratchet to obtain a Brownian particle in an
effective potential. The resulting system is analytically tractable and yet preserves essential dynamical features of
the full model. We first use the QSS reduction to derive an explicit expression for the velocity of a simple two-state
flashing ratchet. In particular, we determine the relationship between perturbations from detailed balance, which
are encoded in the transitions rates of the flashing ratchet, and a tilted-periodic potential. We then perform a
QSS analysis of a pair of elastically coupled flashing ratchets, which reduces to a Brownian particle moving in
a two-dimensional vector field. We suggest that the fixed points of this vector field accurately approximate the
metastable spatial locations of the coupled ratchets, which are, in general, impossible to identify from the full
system.
DOI: 10.1103/PhysRevE.92.042129
PACS number(s): 05.40.Jc, 87.10.−e
I. INTRODUCTION
Noise-driven transport phenomena in spatially periodic
systems are a fundamental feature of many physical processes,
particularly in cell biology [1]. Mathematically speaking, these
processes are often modeled in terms of a Brownian particle
subject to a spatially periodic potential energy environment.
There are at least two distinct mechanisms by which such a
system can be driven away from equilibrium and produce mechanical work. The first, which we will refer to as a Brownian
particle in an effective potential, involves a constant external
force resulting in an effective potential that is tilted periodic,
which means it is more probable that the Brownian particle
escapes the potential barrier on one side [2]. On the other
hand, in the case of a flashing ratchet, the particle undergoes a
set of conformational changes that affect how it interacts with
some underlying periodic structure and perturbations from
equilibrium are encoded in the transitions rates between the
conformational states. In this case, directional transport can
occur when the transition rates do not satisfy detailed balance
[3–5].
Both the flashing and Brownian particle evolving in
effective potentials have rich histories in theoretical studies
of biochemistry and intracellular transport. For example,
these models have been used to gain qualitative insight
into the dynamics of molecular motors [6–10]. Molecular
motors constitute a wide range of nanoscale machines that
are able to convert chemical energy, often in the form of
ATP, into mechanical work by cycling through a number
of conformational states [11]. Examples include flagellar
motors that provide locomotion for single-celled organisms
and cytoskeletal motors that perform a number of tasks
by “walking” along polymerized filaments within cells. A
flagellar motor consists of a set of torque generating stator
proteins, which change conformation in response to an
electrochemical gradient [12], whereas a cytoskeletal motor
like kinesin or dynein has multiple “legs” that attach and detach
to polymerized filaments as a result of chemical reactions [11].
*
bressloff@math.utah.edu
1539-3755/2015/92(4)/042129(10)
The two-state flashing ratchet, which we will examine in
this paper, was first analyzed by Prost et al. [3]. Motivated
by applications to molecular motors, they investigated how
the spatial structure of perturbations from detailed balance
related to the pumping of particles in a flashing ratchet.
Subsequent studies by Prost and co-workers analyzed large
systems of coupled flashing ratchets [6,13–15]. A major
focus of these studies was on self-organization in mean
field models, since systems of flashing ratchets are usually
analytically intractable. As a result, some information about
motor interactions is lost. (For a recent review of other
approaches to modeling cooperative effects in systems of
molecular motors see Ref. [16].)
Analysis of Brownian particles in effective potentials is
more straightforward and they too have a rich history in the
context of molecular motors and intracellular transport. For
example, Peskin and Elston gave theoretical evidence that
motor cargo coupling is elastic [17]. Other more mathematical
studies of Brownian particles explore topics such as stochastic
resonance and coherence in molecular motors [18]. Similar
results for flashing ratchets would be difficult to obtain using
standard tools due to the hybrid nature of these systems.
In this paper we establish a connection between Brownian
particles in effective potentials and flashing ratchets by
carrying out a quasi-steady-state (QSS) or adiabatic reduction,
in which the discrete-state transitions in a flashing ratchet are
absorbed into the deterministic part of the stochastic process,
resulting in an effective tilted-periodic potential for a Brownian
particle. It has been established rigorously that this reduction
is valid for a stochastic hybrid system as the transitions rates
become large [19]. Although the QSS approximation has
previously been used within the context of molecular motors
[20], we are not aware of an instance in which it has been
applied to a flashing ratchet. This is possibly due to the fact
that the fast transition-rate limit does not necessarily hold
for biological molecular motors at the length scales and time
scales associated with single steps. (On the other hand, at the
larger scale of successive runs of a molecular motor, transitions
between runs can be treated as relatively fast [21,22].) Here
we propose that the QSS reduction be used not as a means for
accurately approximating the probability density of the system,
but for obtaining qualitative information about the flashing
042129-1
©2015 American Physical Society
ETHAN LEVIEN AND PAUL C. BRESSLOFF
PHYSICAL REVIEW E 92, 042129 (2015)
The standard analysis of the system (1) proceeds as follow.
First we define the reduced densities and fluxes
k=0
ωoff
ωon
p̂j =
U
FIG. 1. (Color online) Diagram of the two-state flashing ratchet.
ratchet, which would be difficult to obtain otherwise. We first
review the standard analysis of flashing ratchets and show
how the numerical results obtained in Ref. [3] can be derived
analytically from the QSS reduction. We then explore a pair
of elastically coupled flashing ratchets and show that the QSS
reduction allows us to predict the metastable states of the full
system, as well as its mean velocity. The system of two coupled
ratchets has previously been studied numerically in Ref. [10],
although their focus is on the probability current rather than
the identification of metastable states.
√
U (x)
dt + 2DdW (t),
γ
where W (t) is a Wiener process, D is the diffusion coefficient,
and γ is the friction coefficient with Dγ = kB T . Let p1 (x,t)
and p0 (x,t) be the probability that at time t the particle is at
position x and in the attached and detached states, respectively.
The corresponding Chapman-Kolmogorov equation for the
probability vector p(x,t) = (p0 ,p1 ) is
(1a)
(1b)
where the fluxes J0 and J1 are given by
J0 = −D
∂
p0 ,
∂x
J1 = −
U (x)
∂
p1 (x,t) − D p1 (x,t).
γ
∂x
We consider the solutions p(x,t), which are defined on R × R+
and for which the total probability u(x,t) = p0 + p1 satisfies
the normalization condition
u(x,t)dx = 1.
R
Jj (x + nl,t).
n=−∞
U (x)
∂ û(x,t)
Jˆ(x,t) = −
p̂0 (x,t) − D
.
γ
∂x
Unlike the probabilities in the original system, the reduced
probably û will converge to a unique steady state ûSS with
constant probably flux JˆSS . It can be shown that the average
velocity of a particle depends on the probability flux according
to
v=
Jˆ(x)dx = JˆSS .
(2)
0
We begin by reviewing the theory of flashing ratchets;
for a more detailed introduction see Refs. [1,3,6,23]. In the
most general type of flashing ratchet, a particle switches
stochastically between a finite set of discrete states according
to space-dependent transition rates. In each state the motor
evolves as a Brownian particle in a potential. Here we will
restrict out attention to the two-state flashing ratchet, where a
motor can be in either an attached state (k = 1) in which it is
sensitive to a periodic potential U (x) of period or a detached
state (k = 0) in which it diffuses freely in the environment
(see Fig. 1). In the attached state, the location x of the particle
evolves according to the Langevin equation
∂
J0 = −ωon (x)p0 + ωoff (x)p1 ,
∂x
∂
J1 = ωon (x)p0 − ωoff (x)p1 ,
∂x
∞
It follows from linearity that (p̂0 ,p̂1 ) satisfies Eq. (1) and may
be normalized in the interval [0,1]. Next we consider the total
probability û(x,t) = p̂0 + p̂1 , which has the corresponding
probability flux
II. FLASHING RATCHET
∂
p0 +
∂t
∂
p1 +
∂t
pj (x + nl,t), Jˆj =
n=−∞
k=1
dx = −
∞
Upon analyzing this system one finds that unless the detailed
balance condition ωon (x)/ωoff (x) = e−U (x)/kB T breaks down,
the system will have a zero steady-state flux and hence v = 0.
There is a great deal of interest in understanding how the spatial
structure of deviations from detailed balance affect the motor
velocity.
Deviation or perturbation from detailed balance is usually
measured by the quantity
(x) = ωon (x) − ωoff (x)eU (x)/kB T .
(3)
When (x) = 0, one can easily check that the transition rates
satisfy detailed balance or, alternatively, that the probabilities
of occupying the on and off states are given by a Boltzmann
distribution. For simple transitions rates and potential one
can obtain results about how the function (x) relates to
the velocity. In particular, for piecewise linear dynamics
it has been established that in the limiting case in which
perturbations from detailed balance are localized the velocity is
1
a monotonically increasing function of = 0 (x)dx, while
nearly homogenous perturbations yield a unimodal function
for which v → 0 as → ∞ [3].
While it is difficult to say much about the behavior of an
arbitrary flashing ratchet, a Brownian particle evolving in a
single potential is analytically tractable. The flashing ratchet
can be related to an effective potential by carrying out a
QSS reduction. This proceeds as follows. Assume that the
transitions are sufficiently fast (see below) so that at each point
x on very short time scales the probability vector p̂ = (p̂0 ,p̂1 )
converges to the space-clamped steady state ρ = (ρ0 ,ρ1 ) with
ρ0 (x) =
ωoff (x)
,
ωon (x) + ωoff (x)
ρ1 (x) =
ωon (x)
.
ωon (x) + ωoff (x)
Then define the effective velocity ν(x) to be the velocity due to
the potential U (x) weighted by the probability that the motor is
in the attached state, that is, ν(x) = −U (x)ρ1 (x)/γ . Hence in
the limit of fast transitions rates p̂(x,t) evolves as a Brownian
042129-2
QUASI-STEADY-STATE ANALYSIS OF COUPLED . . .
PHYSICAL REVIEW E 92, 042129 (2015)
particle according to
0
x
Uq
0
Ω
0
U (y)ρ1 (y)dy.
0
At steady state Jˆq (x,t) = JˆqSS is constant and solving for ûSS
q
using an integrating factor yields
JˆqSS Z(x)
,
1 − e−{[Uq (y)]y=0 }/kB T
x+
1
eUq (x )/kB T dx .
Z(x) = e−Uq (x)/kB T
D
x
The normalization condition is then used to solve for JˆqSS so
that Eq. (2) becomes
1 − e−{[Uq (y)]y=0 }/kB T
v=
.
(4)
0 Z(x )dx
A necessary condition for the effective potential to generate
motion is that Uq () = Uq (0), that is, Uq (x) is not periodic.
It remains to specify more explicitly the conditions under
which the QSS reduction is valid. In dimensional variables,
if = 10 nm and D = 1 μm2 s−1 [24], the fundamental unit
of time is 10−2 s. A molecular motor such as kinesin makes
around 100 steps per second, suggesting that the transition
rates in Eq. (1) are at least of order 102 s−1 . Therefore,
we will assume 102 s−1 ωon ,ωoff 104 s−1 [24] or, in
dimensionless units, 1 ωon ,ωoff 100. For the moment,
suppose that the dimensionless transition rates are in the range
10–100. This motivates the introduction of a scaling factor such that ωon ,ωoff → ωon /ε,ωoff /ε with the rescaled transition
rates independent of ε. It can then be proved rigorously that
ûSS
q is an exact steady-state solution for the total probability
of the full system in the limit ε → 0 [19]. Moreover, for
0 < 1, the QSS reduction generates an O() correction to
the diffusion coefficient D [21], which can be ignored, and the
QSS solution is still a reasonable approximation. Throughout
the rest of the paper we use the nondimensionalize variables
= 1 and D = 1. We also set the attachment rate to be constant
ωon = ε−1 and express the detachment rate in terms of the
detailed balance relation
0
a l
with a = 1/2 and H (x) the Heaviside function, the potential
itself is asymmetric so even with constant the effective
potential is tilted, as can be seen in Fig. 2(a). On the other hand,
for the sinusoidal potential, constant transition rates would not
produce a tilted potential since U is symmetric. Instead we
have shown a sinusoidal potential with a small phase offset
and it can be seen in Fig. 2(b) that this does indeed produce a
tilted effective potential.
We now derive the relationship between and v as follows.
First, write ρ1 (x) in terms of (x):
ρ1 (x) = [1 + eU (x)/kB T + (x)]−1 .
When the perturbations are nearly uniform, (x) ≈ for
all x. Hence, in the limit → ∞, we have ρ1 (x) → 0 and
thus Uq (x) → 0. It follows that v → 0 when → ∞. [Note
that we could have fixed ωoff (x) = 1, so that ρ1 (x) → 1 and
Uq (x) → U (x) as → ∞ and still v → 0, since U (x) is
periodic.] On the other hand, if perturbations are localized
around an end point, say, x = 0, then in the limit → ∞,
ρ1 (0) = 0 = ρ1 (l) and v = 0. In Fig. 3 we plot the velocity
v of Eq. (4) as a function of . It can be seen that the QSS
velocity’s dependence on the deviation from detailed balance is
0.25
1.8
(a)
x 10-2
(b)
1.35
0.125
homogeneous
localized
9.0
4.5
0.0
0.0 0.2 0.4 _0.6 0.8 1.0
Ω
−1
where we have also rescaled (x) by . This simplifies
matters because the transition rates are completely defined by
U (x) and (x). Figure 2 displays the QSS potential Uq (x) for
U (x) given by a periodic sawtooth function or a sinusoid.
In order for the effective potential Uq (x) to be tilted, the
perturbations from detailed balance must be symmetric. For a
piecewise potential
a l
FIG. 2. (Color online) (a) Periodic piecewise linear potential
(top), perturbations from detailed balance (bottom), and corresponding effective potential Uq . The solid curve for represents a localized
perturbation from detailed balance while the dashed curve represents
an approximately uniform perturbation. We set = 1 throughout. For
the localized case, (x) = (1/a) when a < x < 1 and 0 otherwise.
For the quasiuniform case, (x) = . (b) Corresponding figures for
a sinusoidal potential. The quasiuniform perturbation is now taken to
be (x) = cos(2π x) + .
ωoff (x) = ε−1 [(x) + eU (x)/kB T ],
U (x) = U ∗ [(1 − x/a)H (a − x) + (x/a − 1)H (x − a)],
Ω*
0
velocity v
with an effective potential
x
Uq (x) = −γ
ν(y)dy =
(b)
U*
0
∂ ˆ
∂
û(x,t) +
Jq (x,t) = 0,
∂t
∂x
Uq (x)
∂
Jˆq (x,t) = −
û(x,t) − D û(x,t)
γ
∂x
ûSS
q (x) =
(a)
U
0.0
0.0
2.5
_
Ω
5.0
FIG. 3. Plot of flashing ratchet velocity v as a function of using piecewise linear transitions rates and potentials (see Fig. 2)
for (a) ε = 0.01 and (b) ε = 1. Solid (dashed) curves correspond to
a quasiuniform (localized) perturbation (x). The solutions of the
flashing ratchet model are given by the gray curves and the solutions
of the reduced model are given by the black curves. The parameter
values are D = 1, = 1, and a = 0.4.
042129-3
ETHAN LEVIEN AND PAUL C. BRESSLOFF
PHYSICAL REVIEW E 92, 042129 (2015)
in qualitative agreement with that of the full system even when
= 0.1. Consistent with the findings of Ref. [3], the velocity
is a monotonically increasing function of for a localized
perturbation and a unimodal function of for a quasiuniform
perturbation.
⎛
(1)
(2)
(x1 ) − ωon
(x2 )
−ωon
(2)
⎜
(x
)
ω
on 2
A(x,t) = ⎜
(1)
⎝
(x1 )
ωon
0
ω(1)
off
vs(1) (x)
= −κ(x1 − x2 − z) + f1 (x1 )s1 ,
while V2 has diagonal entries
vs(2) (x) = κ(x1 − x2 − z) + f2 (x2 )s2
with
fi (xi ) = −Ui (xi )/γ .
Observe
that
vs (x) =
[vs(1) (x),vs(2) (x)]T is the gradient of a two-dimensional
scalar potential. Hence, in any given state we can think of
the motor pair as a single Brownian particle moving in the
two-dimensional potential
Ustot (x) = 12 κ(x1 − x2 − z)2 + U1 (x1 )s1 + U2 (x2 )s2 ,
with vs (x) = −∇Ustot (x)/γ . In the remainder of this paper
we often refer to x as a single particle evolving in a
two-dimensional spatial domain. This interpretation is advantageous because it relates the coupled model to a twodimensional flashing ratchet.
ω(1)
off
ω(1)
on
ω(1)
on
ω(2)
off
ω(2)
on
s = 11
s = 10
FIG. 4. (Color online) Diagram of the discrete internal state
space for two coupled flashing ratchets.
so that the elastic energy of the system is
Uspring (x) = 12 κ(x1 − x2 − z)2 ,
where z is the equilibrium position of the spring.
The discrete internal states of the system are elements of
the set S = {00,01,10,11} (see Fig. 4). We start by writing
down the Chapman-Kolmogorov equation for the probability
ps (x,t) that the system is in state s ∈ S with motor position
x = (x1 ,x2 ) at time t. We use si ∈ {0,1} to denote the state
of the ith motor; for example, in the state s = 10, motor 1 is
in state s1 = 1. Given the above assumptions, the probability
vector p(x,t) = (p00 ,p01 ,p10 ,p11 ) evolves according to
∂
∂
∂
1
p=−
V1 p −
V2 p + D∇ 2 p + A(x,t)p,
∂t
∂x1
∂x2
ε
where
(2)
ωoff
(x2 )
(2)
(1)
−ωoff (x2 ) − ωon
(x1 )
0
(1)
(x1 )
ωon
and V1 has diagonal entries
s = 00
ω(2)
on
III. PAIR OF ELASTICALLY COUPLED
FLASHING RATCHETS
We next consider a system of two elastically coupled
flashing ratchets with possibly different dynamical properties.
While many studies have analyzed collective behavior in mean
field models of coupled ratchets [6,13–15], there are very few
analytical results concerning the effects of coupling on small
numbers of flashing ratchets. One problem is that for small
numbers of coupled particles the assumption that coupling is
rigid breaks down, which greatly increases the complexity of
the calculations. Note that each ratchet can be interpreted as a
molecular motor [25,26] or as the head of a single two-headed
motor [10].
Extending the ideas applied to the single flashing ratchet,
we will show how the QSS reduction can be used to gain insight
into the dynamics of elastically coupled flashing ratchets,
which are analytically intractable for even the simplest choices
of rate functions and potentials. In this model each motor
is associated with an -periodic potential Ui (xi ) and rate
(i)
(i)
(xi ) and ωoff
(xi ) describing the rate of attaching
functions ωon
and detaching from the potential, respectively; xi is the spatial
position of the ith motor. One simplifying assumption is that
the transition rates depend only on the position of a single
motor, hence strain-dependent unbinding has been neglected
[25]; this assumption has also been applied to two coupled
heads of a single motor [10]. We take the tethers to be Hookean
ω(2)
off
s = 01
(1)
ωoff
(x1 )
0
(1)
(2)
−ωoff
(x1 ) − ωon
(x2 )
(2)
ωon (x2 )
⎞
0
(1)
⎟
ωoff
(x1 )
⎟
(2)
⎠
ωon
(x2 )
(1)
(2)
−ωoff (x1 ) − ωon (x2 )
To illustrate the utility of the QSS reduction we first
consider a system for which the center of mass has zero
velocity. (Nonzero velocities are considered later.) To do this
we consider the simple case in which each particle moves in
the same sinusoidal potential U and has the same sinusoidal
perturbation from detailed balance [see Fig. 2(b)]. The only
difference is that the sinusoidal functions of the first motor
are out of phase by 1/4 with respect to the second particle. In
other words, U2 (x) = U1 (x − 1/4) and 2 (x) = 1 (x − 1/4).
Shifting the phase of the potentials ensures that the advection
terms for the two particles do not share the same zeros. As
we will soon see, this makes the identification of metastable
spatial positions nontrivial and motivates the application of the
QSS reduction.
Let us begin by noting that in the limit κ → ∞ the
effective Brownian particle moves along the line x2 = x1 ,
hence the distance between the motors is fixed and they
move synchronously in one dimension. Decreasing κ should
042129-4
QUASI-STEADY-STATE ANALYSIS OF COUPLED . . .
x [10nm]
3.0
(a)
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
0
PHYSICAL REVIEW E 92, 042129 (2015)
1000
2000
3000
4000
5000
1000
2000
3000
time t [ms]
4000
5000
x [10nm]
2 (b)
0
-2
-4
-6
0
FIG. 5. Stochastic trajectories x = x1 ,x2 of the two particles
(indicated by black and gray lines, respectively) with deterministic
dynamics corresponding to (a) a hard spring (κ = 8) and (b) a soft
spring (κ = 0.1). Other parameters are D = 1, z = 0, and ε = 1.
Trajectories were generated by running a forward Euler method on
the SDE for the current state and calculating the time until the next
state transition. If the time is less than the time step dt = 0.01, then
a random number is generated to select a state transition.
introduce metastable points off the line x1 = x2 , which
increases the number of stable configurations we observe and
hence increases the variance in motor position. This effect can
be seen in numerical simulations (see Fig. 5). Note that the
system seems to spend most of its time with the particles close
together, although not at the exact same spatial position. This
behavior is reminiscent of the common modeling approach in
which space is discretized and the various motor configurations
are characterized by the elastic energy between the two motors
[25–27]. In contrast to previous work on coupled ratchets, we
do not assume that the discrete energy states are located at
regular intervals on a lattice; instead we want to determine
their location by analyzing the flashing ratchet system.
Figures 6 and 7 show the four vector fields vs (x), s ∈
{00,01,10,11}, for two different values of the spring constant κ
and z = 0. It is tempting to relate these figures to the different
transport regimes observed in Fig. 5, but it is not immediately
clear what information about the system can be extracted
from these vector fields. In the absence of state transitions,
fixed points of these vector fields are metastable states of
the corresponding continuous stochastic process for that state.
However, in the stochastic hybrid system the particle does not
necessarily have time to converge to one of these metastable
locations and once it arrives it may transition to a state where
the same location is not metastable. In this example, we have
explicitly chosen potentials that do not share the same fixed
points. Therefore, it is not clear whether or not any metastable
states exist and if they do, whether or not they coincide
with those of any of the individual states. In fact, even if
all potentials share the same fixed points and their basins of
attraction coincide, it is possible that the location of the fixed
points may not correspond to a metastable location in the full
system [28].
In order to make meaningful statements about the dynamics
of the coupled ratchet system, one must determine how to
incorporate information about the transition rates and establish
how the individual potentials relate to the dynamics of the
full system. For problems in which noise is weak, a WKB
approximation can be used to construct a quasipotential,
which accurately approximates the evolution of a particle
in the stochastic hybrid system [29]. Unfortunately, the
WKB approximation can be difficult to construct and is not
appropriate for the flashing ratchet where diffusion acts as a
strong source of noise. We now demonstrate how the QSS
reduction can be used to obtain an effective two-dimensional
system that contains much of the information about the full
model we are interested in.
As in the single ratchet case, when the switching is relatively
fast (ε → 0), the probability density will converge to a unique
steady-state density ρ at each x, which satisfies Aρ = 0. In
this limit we derive a scalar Fokker-Planck (FP) equation for
the total probability
ps (x,t),
u(x,t) =
s∈S
which is subject to the normalization condition
u(x,t)dx = 1.
R2
The effective velocity for u is given by weighting each velocity
vector vs = (vs(1) ,vs(2) )T by the probability that the system is in
state s, so we have
ν(x) =
ρs (x)vs (x).
s∈S
Neglecting O(ε) terms, the FP equation for u is
∂u
= −∇ · [ν(x)u] + D∇ 2 u.
(5)
∂t
After some computation we find that the QSS velocity of the
ith motor is given by
νi (x) = (−1)i κ(x1 − x2 − z) + fi (xi )σi (xi ),
(6)
where
σi (xi ) =
(i)
(xi )
ωon
(i)
(i)
ωon
(xi ) + ωoff
(xi )
.
(7)
Following the analysis of the single ratchet, let 1 (x) and
2 (x) measure the perturbations from detailed balance of the
two motors. Specifically,
(i)
(i)
i (x) = ωon
(x) − ωoff
(x)eUi (x)/kB T .
(i)
Taking ωon
(x) = 1, we have
σi (x) = [1 + eUi (x)/kB T + i (x)]−1 .
In general, it is more difficult to interpret the dynamics
of a particle in a two-dimensional vector field resulting
from the QSS of the coupled ratchets because ν(x) is not
042129-5
ETHAN LEVIEN AND PAUL C. BRESSLOFF
PHYSICAL REVIEW E 92, 042129 (2015)
x2
(a) 1
(b) 1
0
−1
−1
0
0
x1
(c) 1
x2
−1
−1
1
1
0
x1
1
(d) 1
0
−1
−1
0
x1
0
0
x1
−1
−1
1
FIG. 6. (Color online) Plots of the vector field vs (x) (dark lines) and nullclines (light lines) for motor 1 (solid lines) and motor 2 (dashed
lines) in (a) the fully attached state (s = 11), (b) the fully detached state (s = 00), (c) the particles-one attached state (s = 10), and (d) the
particles-two attached state (s = 01). Both particles move in the same sinusoidal potential modulo a phase shift. The parameter values are
D = = 1, = 10, κ = 8, and z = 0.
necessarily the gradient of a potential. However, examining
the vector field ν(x) yields insight into qualitative features of
the system, as we know that for small ε the QSS provides
an accurate approximation to the full system. For ε → 0
the stochasticity resulting from state transitions is absorbed
into the deterministic part of the system. Therefore, provided
we have chosen parameter values such that ν(x) is not at a
bifurcation point, there exist small but nonzero values of ε for
which we expect the full system and the QSS system to share
the same metastable states.
A visual comparison of ν(x) with the stochastic trajectories
of the full system reveals that the QSS accurately predicts the
metastable states for a range of ε values. We show this in Fig. 8,
where it can be seen that even for relatively large values of ε the
particle fluctuates near the fixed points of ν(x). Note that the
initial condition for the stochastic trajectories is in the basin of
attraction for a fixed point of v11 but the stochastic trajectories
evolve away from this fixed point towards fixed points of ν. In
Fig. 9 we show the same for κ = 0.1. In this case v11 and ν
nearly agree but the QSS was needed to establish this fact. In
both cases the QSS vector fields appears to represent a small
perturbation of the exact dynamics. We would expect that if
there exists a quasipotential, such as one derived from a WKB
approximation, the QSS would approximate the corresponding
gradient vector field. Our numerical simulations for other
transitions rates indicate that the QSS reduction is a reliable
method for determining the metastable states of the coupled
ratchet system.
Thus far, for the coupled system we have only demonstrated
that the QSS approximation accurately predicts the behavior
of the system over short time scales. While this is indeed
interesting in its own right, the results for the single motor,
and specifically those presented in Fig. 3, suggest that one
can also use the QSS approximation to determine the mean
velocity of the coupled ratchets as a function of biophysical
parameters. Let v c denote the average velocity of the center
of mass for the coupled system. In Fig. 10(a) we show
numerical results comparing the mean velocity of the full
system with that of the SDE corresponding to Eq. (5). For
the sake of illustration, we consider asymmetric transition
rates and homogenous perturbations from detailed balance.
It can be seen that the velocity of the center of mass is a
monotonically increasing function of spring stiffness κ in
both cases and there is reasonable quantitative agreement.
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QUASI-STEADY-STATE ANALYSIS OF COUPLED . . .
PHYSICAL REVIEW E 92, 042129 (2015)
x2
(a) 1
(b) 1
0
−1
−1
0
0
x1
(c) 1
x2
−1
−1
1
1
0
x1
1
0
x1
1
(d) 1
0
−1
−1
0
x1
0
0
x1
−1
−1
1
FIG. 7. (Color online) Same as in Fig. 6 with κ = 0.1.
x2
(a) 1
(b) 1
0
−1
−1
0
ε=1
ε = 0.1
ε = 0.01
0
x1
1
−1
−1
FIG. 8. (Color online) (a) Stochastic trajectories of the full system superimposed on the QSS vector field ν(x) for various values of ε.
The nullclines are shown as thicker (red) curves for motor 1 (solid lines) and motor 2 (dashed lines). The parameters used are κ = 8, D = 1,
and = 10. The stochastic trajectories were generated using the same algorithm as in Fig. 5 for T = 10 ms with dt = 0.001 ms and all
three simulations are with the initial condition x(0) = (−0.75,0.5). Note that to make the figure readable T was chosen so that we only see
convergence to a single metastable state. This is in contrast to Fig. 5, where the transitions between metastable states can be observed. (b) The
QSS nullclines (dark thick curves) superimposed on the vector field v11 (x) and nullclines (thick light curves) for the fully attached state of the
full system.
042129-7
ETHAN LEVIEN AND PAUL C. BRESSLOFF
PHYSICAL REVIEW E 92, 042129 (2015)
x2
(a) 1
(b) 1
0
−1
−1
0
ε=1
ε = 0.1
ε = 0.01
−1
−1
1
0
x1
1
0
x1
FIG. 9. (Color online) Same as in Fig. 8 with κ = 0.1.
Figure 10(b) shows convergence to the QSS velocity as ε → 0
for fixed κ. Our results are consistent with a previous study
of the strain-independent coupled ratchet model [10]. Note
that in models of coupled molecular motors that include
strain-dependent unbinding [25], one finds that increasing κ
reduces the velocity.
We now show how one can derive an expression for the
velocity of the center of mass by utilizing the symmetries of
Eq. (5) to obtain a system that is periodic in the horizontal
(i)
(i)
direction. First, given the assumptions on ωon
, ωoff
, and fi (xi ),
it follows that fi (xi )σi (xi ) is periodic in xi . Hence, in the
absence of elastic coupling the original system is doubly
periodic with respect to the transformations
for arbitrary integers (n1 ,n2 ). In the presence of elastic
coupling this symmetry reduces to
velocity vc
1.4
1.2 (a)
1.0
0.8
0.6
0.4
Full model
QSS
0.2
0.0
-0.2
0 2
4 6 8 10
spring stiffness κ
x2 → x2 + n,
the νi are invariant under the transformation
ξ1 → ξ1 + n, ξ2 → ξ2 .
Therefore, consider the change of variables x → ξ = (ξ1 ,ξ2 ).
In matrix notation the appropriate transformation is given by
1 −1
R=
1
1
∂ ũ
= −∇ · [ν̃(ξ )ũ] + D∇ 2 ũ,
∂t
where ũ(ξ ,t) = u(R−1 ξ ,t) and
We simplify notation by dropping the tildes in the remainder
of this section. Note that ξ1 is proportional to the center
of mass, while ξ2 measures the displacement of the spring
from equilibrium. It is also worth noting that after this
transformation we have the useful relation
Full model
QSS
0.8
0.6
0.4
0.2
0
(8)
ν̃(ξ ) = Rν(R−1 ξ ).
1.6
(b)
1.4
1.2
1.0
1
1
ξ1 = √ (x1 − x2 ), ξ2 = √ (x1 + x2 ),
2
2
so that ξ = Rx. Now Eq. (5) can be rewritten as
x1 → x1 + n1 , x2 → x2 + n2 x1 → x1 + n,
i.e., n1 = n2 . Under the latter symmetry νi (x) is invariant.
Hence, defining
ν2 (ξ ) = −2κ(ξ2 − z) − ν1 (ξ )
and, as we would expect,
lim v2 (ξ ) > 0,
0.2 0.4 0.6 0.8 1.0
ε
FIG. 10. (Color online) (a) Average velocity v c as a function of
κ for the full model and QSS. Simulations for the full and reduced
system were performed using Euler’s method for the SDE, taking
into account switching in the full system. Here we have used the
piecewise linear potentials with a /4 phase offset and homogenous
perturbations from detailed balance shown in (a). The parameters
used are D = 1, a = 0.1, and ε = 0.001. (b) Average velocity v c as
a function of ε showing convergence to the QSS velocity as ε → 0.
The same parameter values are used and κ = 8.
ξ2 →−∞
lim v2 (ξ ) < 0.
ξ2 →∞
We introduce the reduced density
û(ξ ,t) =
∞
u(ξ1 + n,ξ2 ,t).
n=−∞
Linearity and the periodicity of ν guarantee that û satisfies
Eq. (8) on the strip [0,] × R, subject to the periodic boundary
conditions
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û(0,ξ2 ,t) = û(,ξ2 ,t).
QUASI-STEADY-STATE ANALYSIS OF COUPLED . . .
PHYSICAL REVIEW E 92, 042129 (2015)
The fact that the domain of ξ1 is finite along with the limiting
behavior of ξ2 guarantees that there is a time-independent
steady-state solution ûSS (ξ ) = û(ξ ,t). We consider the probability flux
To summarize, we have shown that the QSS reduction is a
useful tool for obtaining qualitative information about flashing
ratchets. When applied to a single-particle flashing ratchet, the
QSS reduction yields a single effective potential describing the
motion of the system, relating two fundamental mechanisms
for generating noise-induced transport in periodic media.
The QSS reduction converts a spatially periodic stochastic
hybrid system, in which nonequilibrium perturbations in
the discrete dynamics give rise to directed transport, to a
continuous-time stochastic process where information about
the nonequilibrium perturbations is contained in an aperiodic
forcing term. The latter system is significantly easier to study.
We derived the relationship between mean particle velocity
and perturbations from detailed balance, namely, that the
velocity is a monotonic function of the amplitude of the
perturbations when they are localized and unimodal when they
are homogenous. While this relationship has previously been
shown for specific transitions rates, we are not aware of a more
general derivation.
In general, identifying metastable states in stochastic hybrid
systems is a difficult problem. For the single on-off flashing
ratchet, it is clear that the metastable states are located at the
minima of the nontrivial potential and the QSS accurately
preserves this. For a system of two coupled flashing ratchets
the situation is more complicated and an inspection of the
individual potentials does not allow one to identify the
metastable states. However, such states can be determined
from the effective vector field obtained using a QSS reduction.
The existence and distribution of metastable states and the
resulting asynchrony in the stepping of the two motors or,
alternatively, a single motor consisting of two heads, strongly
depend on the elasticity of the tether. This is consistent with
previous studies of elastically coupled motors, in which space
is discretized [25–27,30]. We have also presented numerical
results that suggest that the QSS preserves the velocity of the
system and derived an expression for the velocity of the center
of mass.
There are a number of possible extensions of our work.
For example, one could extend the analysis of two coupled
ratchets to larger systems of collective transport that go beyond
mean field theory. In mean field models one often takes
the coupling between particles to be rigid. However, this
simplification breaks down for a finite number of particles
leading to analytically intractable systems. As a result, few
studies have considered how elastic coupling effects collective
transport by many molecular motors. The QSS reduction
of a system of N two-state flashing ratchets will yield a
Brownian particle evolving in an N -dimensional domain. In
order to extend our analysis to systems with N > 2, one
would need to specify how the motors are coupled. For
example, nearest-neighbor coupling between motors would
likely yield different dynamic behavior than coupling to a
single cargo. Hence extensions of our analysis require one to
be more specific about the corresponding physical or biological
system of interest. It would also be of interest to carry out a
more general mathematical investigation of the relationship
between stochastic hybrid systems and the continuous system
resulting from the QSS reduction. Obtaining a quasipotential
from a WKB approximation is useful for escape problems
in stochastic hybrid systems in which noise is weak and the
interval state dynamics are deterministic. However, when there
is a strong diffusion within the individual states, the QSS may
be sufficient to identify the import features of the dynamics.
We are interested in what features of a stochastic hybrid system
are preserved and in obtaining bounds on the values of ε for
which the dynamics are equivalent up to homotopy.
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Ĵ(ξ ,t) = ν(ξ )û − D∇ û.
At steady state the horizontal component of the flux satisfies
J1SS (ξ ) = ν1 (ξ )ûSS (ξ ) − D
∂ SS
û (ξ )
∂ξ1
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(9)
(10)
[0,]×R
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IV. DISCUSSION
042129-9
ETHAN LEVIEN AND PAUL C. BRESSLOFF
PHYSICAL REVIEW E 92, 042129 (2015)
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